1. F PAT 1 ( . .53)
1. ก F p q ˈ F
F F F ˈ
1. (p ⇒ q) ∨ p
2. (∼ p ∧ p) ⇒ q
3. [(p ⇒ q) ∧ p] ⇒ q
4. (∼ p ⇒ q) ⇔ (∼ p∧∼ q)
2. F F ก F
1. F ก F {−1, 0, 1}
F ˈ∀x∃y[x2 + x = y2 + y]
2. F ก F ˈ
F ˈ∃x[3x = log3x]
3. F ก F ˈ
F ∀x∃y[(x > 0 ∧ y ≤ 0) ∧ (xy < 0)]
∃x∀y[(xy < 0) ⇒ (x ≤ 0 ∨ y > 0)]
4. F ก F ˈ
F ∀x[x > 0 ⇒ x3 ≥ x2]
∃x[(x ≤ 0) ∧ (x3 < x)]
3. F A = {1, {1}} P(A) ˈ F A
F F
1. ก F ก 3P(A) − A
2. ก P(P(A)) F ก 16
3. {{1}} ∈ P(A) − A
4. {∅,A} ∈ P(A)
1
ˆ F F F

2. 4. ก F A = {x ∈ R x2 − 6x + 9 ≤ 4}
R
F F ก F
1. A = {x ∈ R 3 − x > 4}
2. A ⊂ (−1,∞)
3. A = {x ∈ R x ≤ 7}
4. A ⊂ {x ∈ R 2x − 3 < 7}
5. ก F x ˈ F F ก 1y1 = f(x) = x+ 1
x− 1
y2 = f(y1), y3 = f(y2),.....
n = 2, 3, 4, .....yn = f(yn− 1)
F ก F Fy2553 + y2010
1. x −1
x +1
2. x2 +1
x−1
3. x2 +1
2x
4. 1 +2x−x2
x− 1
6. F f g ˈ ˆ กF ก
f(x) = x −1
x2 − 4
g(x) = f(x) − x − 1
F F
ก. Dg = (2, ∞)
. F x > 0 F g(x) = 0 1 F F
F F ก F
1. ก. ก . ก
2. ก. ก F .
3. ก. F . ก
4. ก. .
2
ˆ F F F

3. 7. ก F x ˈ
F sin x + cos x = a sin x − cos x = b
F F sin 4x F ก F F
1. 1
2
(a3b − ab3)
2. 1
2
(ab3 − a3b)
3. ab3 − a3b
4. a3b − ab3
8. ก F ก ˈ 25x2 + 21y2 + 100x − 42y − 404 = 0
F F F ก F (−3, 1 + 8 )
ก ก F F
1. 5y2 − 4x2 − 10 8 y − 32x − 25 = 0
2. 3y2 − 2x2 − 6 8 y − 8x + 15 = 0
3. y2 − 4x2 − 2y − 16x − 19 = 0
4. y2 − 7x2 − 2y − 28x − 28 = 0
9. ˈ ABCDA(−3,1) B(1, 5) C(8, 3) D(2,−3)
F F
1. F AB ก F DC
2. ก F AB ก DC F ก F10 2
3. ก ก A F F C D F F ก F9 2
2
4. ก ก B F F C D F F ก F9
2
10. ก F x y ˈ ก y ≠ 1
F F x F F ก F Flogy2x = a 2y = b
1. 1
2
(log2b)a
2. 2(log2b)a
3. a
2
(log2b)
4. 2a(log2b)
3
ˆ F F F

4. 11. ก ˈ F F72x + 72 < 23x+ 3 + 32x +2
1. (log87 , log98)
2. (log98 , log89)
3. (log89 , log78)
4. (log910 , log89)
12. F ก ˈ ก

1
4


x
+ 

1
2


x −1
+ a = 0
F F a ˈ F F F F F
1. (−∞,−3)
2. (−3,0)
3. (0, 1)
4. (1, 3)
13. ก F f

x
x− 1

 = 1
x x ≠ 0 x ≠ 1
F F F ก F F0 < θ < π
2
f(sec2θ)
1. sin2θ
2. cos2θ
3. tan2θ
4. cot2θ
14. F ˈ ก F กa b
p ˈa = i + 1
2
j − 3pk b = − 2pi + 2j + pk
F กก F ก 3 Fa b b
F p F F F F
1. (−3,−3
2
)
2. (−3
2
, 0)
3. (0, 3
2
)
4. (3
2
, 3)
4
ˆ F F F

5. 15. ก F ABC ˈ A(0, 0) B(2, 2) ˈ C(x, y)
ˈ (quadrant) 2 F F AC F ก F BC F
ABC F F ก 4 F F C F F F F
1. x − y + 4 = 0
2. 4x + 3y − 1 = 0
3. 2x − y − 3 = 0
4. x + y − 5 = 0
16. F ˈ FZ1, Z2, Z3,.....
Z1 = 0,
n = 1, 2, 3, .....Zn+ 1 = Zn
2
+ i i = −1
F F F ก F FZ111
1. 1
2. 2
3. 3
4. 110
17. ก ก F ก F F3 + 11
4
+ 33
16
+ ∧+3n +2n − 2
4n−1
+ .....
1. 20
3
2. 29
3
3. 31
3
4. 40
3
18. ก F R F ˈ ˆ กFf : R → R g : R → R
F F ก F Ff(x) = 3x
2
3 , g(1) = 8 g (1) = 2
3
(fog) (1)
1. 1
3
2. 2
3
3. 1
4. 4
3
5
ˆ F F F

6. 19. ก F 13 4 F S, M, L
XL F กก F 3 F ก F ˈ F
ก 2 F ก F F
1. 72
425
2. 72
5525
3. 3
221
4. 3
22100
20. ก F S ˈ ʽ A, B ˈ ก F S
F F
ก. P(A) = P(A ∩ B) + P(A ∩ B )
. F P(A) = 0.5 , P(B) = 0.6 P(A ∪ B ) = 0.7
F P(A − B) = 0.4
F F ก F
1. ก. ก . ก
2. ก. ก F .
3. ก. F . ก
4. ก. .
21. ก F F F F ก 40 F
ก F 35 ก F
50 F ก F ก ก F F
1. 3 : 2
2. 2 : 3
3. 2 : 1
4. 1 : 2
22. ก F A = 7(77) , B = 777 , C = 777 D = (777)7
F F ก F
1. B < A < C < D
2. B < C < A < D
3. C < B < D < A
4. C < A < D < B
6
ˆ F F F

7. 23. F ก F " PAT"
16325, 34721, 12347, 52163, 90341, 50381
F F ˈ PAT
2564, 12345, 854, 12635, 34325, 45026
F F ˈ " PAT"
1. 75401
2. 13562
3. 72341
4. 83051
24. F N
ก F a ∗ b = ab a,b ∈ N
F F
a,b,c ∈ N
ก. a ∗ b = b ∗ a
. (a ∗ b) ∗ c = a ∗ (b ∗ c)
. a ∗ (b + c) = (a ∗ b) + (a ∗ c)
. (a + b) ∗ c = (a ∗ c) + (b ∗ c)
F F ก F
1. ก 2 F . .
2. ก 2 F . .
3. ก 1 F .
4. ก. . . . ก F
7
ˆ F F F

8. 25. F F F 5 A, B, C, D E
A ก F "C D ก ก"
B ก F "A C ˈ "
C ก F "D ก ก"
D ก F "E ก ก"
E ก F "B ก ก"
ก F ก F F F F F F F ˈ F ˈ
1. A, B, D C E
2. B D A C
3. A, B C D E
4. B E A C
26. ก F A, B C ˈ
F n(A ∪ B ∪ C) = 91 , n(A ∩ B ∩ C ) = 11,
n((B − A) ∩ (B − C)) = 15 , n(A ∩ B ∩ C) = 20
n(C) = 59n((A ∩ B) ∪ (A ∩ C) ∪ (B ∩ C)) = 47
F F ก Fn(A ∩ B ∩ C)
27. F S = {x ∈ R 3x + 1 + x − 1 = 7x + 1 }
R
F ก ก S F ก F
28. F A ˈ ก F F ก F F ก 10
B ˈ ก F F ก F F ก 10
C ˈ ˆ กF ˈ ˆ กF Ff : A → B
. . . a f(a) F F ก 1 ก F กa ∈ A
C F ก F
29. F ˈ กα β tan α = a
b
F cos


arcsin



a
a2 +b2





 + sin


arccos



a
a2 + b2





 = 1
F F F ก Fsin β
8
ˆ F F F

9. 30. F F ก Fcos 36 − cos72
sin 36 tan 18 + cos36
31. F A B ˈ ก F 2 × 2
2A − B =



−4 −4
5 6


 A − 2B =



−5 −8
4 0



F F ก Fdet(A4B−1)
32. F x, y, z w F ก ก



1 0
−1 w






x −1
0 y


 =



2y −1
z 2






1 0
−1 w



F F ก F4w − 3z + 2y − x
33. F ˈ ก F กu , ν w
a, b, c d ˈu = i + 2j + 3k , ν = 2i − dj + k , w = ai + bj + ck
F q , r ˈu ⋅ w = 2 , u ⋅ (ν + w) = 3 , ν + w = i + qj + rk
กw −2
3
i + 1
2
j + 1
3
k
F F a + 4b + 2c F ก F
34. F ˈ F (conjugate)Z1 Z2 Z2 Z2
F F5Z1 + 2Z2 = 5 Z2 = 1 + 2i i2 = −1
F F ก F5Z1
−1
35. F ˈ{an}
ก ก nan = 2+ 4+6 +K+ 2n
n2
F F F ก Fn→ ∞
lim an
36. ก F n = 1, 2, 3, .....Sn =
n
k =1
Σ



1
k (k+ 1)+k k+1



F F ก Fn→ ∞
lim Sn
9
ˆ F F F

10. 37. ก F a b ˈ f ˈ ˆ กF ก
f(x) =







x3 −3x− 2
x −2
, x < 2
a − b , x = 2
x2 + ax + 1 , x > 2
F f ˈ ˆ กF F F
F F ก Fa2 + b2
38. ก F R F ˈ ˆ กFf : R → R
ก x f(1) = 5f (x) = 3 x + 5
F F F ก F
x→ 4
lim
f(x2)− 2
f(x)
39. ก F R F ˈ ˆ กFf : R → R
ก x F F F y = f(x)f (x) = 6x + 4
(2, 19) F ก 19 F F f(1) F ก F
40. ก F A = {0, 1, 2, 3, 4} ก F F ก F 300 F ก
A F ก F ก F ก F
41. ก ก 7 ก F ก
ก ก ก 4 ก F 7 F ก F
ก F ก F ก F ก
F
42. F ก ก F F ก 72
( ก ) F ก 600 F ก ก 1 F 60 F
F ˈ 70 F F F ก F
43. กก ก ก ก F 4 2 ก F ก
ก F ก F ก 2 F ก ก
4 45, 46 6 ก ก F ก
ก 4 F ก F
44. ก ก F ก F F F F 700
ˈ F F 4 F F F 400 ˈ F
F F ก F ก F F−2
10
ˆ F F F

11. 45. F ʾ F 4 ก F 4 F
F 20 ʾ ก
( F F 1 F 2
F 3 F 4
ก F F 5 F F 6
F F 7)
46. ก ก 221 ก ก ก 260 ก F ก F
ก ก ก ˈ ก ก
(1) F ก ก
(2) ก F ก F ก
F F ก F ก ก ก F ก F F Fกก
47. ก F R ˈ
F ˈ ˆ กFf : R → R g : R → R
ก ก ก ⊗⊗⊗⊗ f g
⊗⊗⊗⊗(f g)(x) = f(g(x)) − g(f(x))
ก x
F g(x) = 2x + 1 ก xf(x) = x2 − 1
F (f ⊗⊗⊗⊗ g)(1) F ก F
48. F a, b, c, d ˈ ก F ก F 4 ก dcba F ก 9 F
abcd F b F ก F
11
ˆ F F F

12. 49. F
F ก 1, 2, 3,....., 11 F F 1 F
ก F ก 43 ก F ก 28
x F F ก F
50. ก 2, 3, 4, 5, 6,..... F
1 9 17 ...
2 2 8 10 16 ...
3 3 7 11 15 ...
4 4 6 12 14 ...
5 5 13 ...
2400 F F
*************************
x
12
ˆ F F F

13. F PAT 1 ( . .53)
F 1 4
1 ∼ p ∨ q ∨ p ≡ (∼ p ∨ p) ∨ q ≡ T ∨ q ≡ T
2 F → q ≡ T
3 ≡ ∼ ((p → q) ∧ p) ∨ q
≡ ∼ (p → q)∨∼ p ∨ q
≡ ∼ (p → q) ∨ (p → q) ∼ A ∨ A ≡ T
≡ T
4 ≡ ∼ (∼ p) ∨ q ↔ ∼ (p ∨ q)
≡ (p ∨ q) ↔ ∼ (p ∨ q) ≡ F
A ↔ ∼ A ≡ F
F 2 3
F 1 x = − 1, y = − 1 (−1)2 + (−1) = (−1)2 + (−1) T
x = 0, y = 0 02 + 0 = 02 + 0 T
x = 1, y = 1 12 + 1 = 12 + 1 T
F 2 F ก Fy = 3x y = log3x
∴ F x F 3x = log3x
F 3 ก ∼ ∀x∃y[p ∧ q ∧ r]
≡ ∃x∀y[∼ p∨∼ q)∨∼ r]
≡ ∃x∀y[r → (~p∨∼ q)]
≡ ∃x∀y[xy < 0 → (x ≤ 0 ∨ y > 0)]
F 4 ∼ ∀x[p → q] ≡ ∼ ∀x[∼ p ∨ q] ≡ ∃x[p∧∼ q] ≡ ∃x[x > 0 ∧ x3 < x2]
y
x
1
ˆ F F F

14. F 3 4
ก F FA = {1,{1}} P(A) = {∅,{1},{{1}},{1,{1}}}
P(A) − A = {∅,{{1}},{1,{1}}}
ก 1 ก n(P(A) − A) = 3
ก 2 ก n(P(P(A)) = 22n(A)
= 222
= 16
ก 3 ก {{1}} ∈ P(A) − A
ก 4 {∅,A} = {∅,{1,{1}}} ∉ P(A)
F 4 1
A (x − 3)2 ≤ 4
x − 3 ≤ 4 → − 4 ≤ x − 3 ≤ 4
−1 ≤ x ≤ 7
A = [−1,7] → A = (−∞,−1) ∪ (7,∞)
1 A 3 − x > 4 → x − 3 > 4
x − 3 > 4 x − 3 < − 4
x > 7 x < − 1
A = (−∞,−1) ∪ (7,∞)
F 5 2
y2 = f

x +1
x −1

 =


x+1
x−1

 +1


x+1
x−1

 −1
=
x+1 +x−1
x −1
x+1 −(x−1)
x −1
= 2x
2
= x
y3 = f(y2) = f(x) = x+1
x−1
y4 = f(y3) = f

x+ 1
x− 1

 = x
F y y F = x= x+ 1
x− 1
,
∴ =y2553 + y2010 = x+1
x−1
+ x x +1+ x2 −x
x −1
= x2 +1
x−1
2
ˆ F F F

15. F 6 4
g(x) = x−1
x2 −4
− x − 1
Dg
x−1
x2 −4
≥ 0 x − 1 ≥ 0
(x −1)
(x −2)(x+2)
≥ 0 ∩ x ≥ 1
∴ ก.Dg = {1} ∪ (2,∞)
g(x) = 0 x−1
x2 −4
= x − 1 ⇒ x− 1
x2 −4
= x − 1
∴x = 1 1
x2 −4
= 1 → x2 − 4 = 1 → x2 − 5 = 0 → x = 5 ,− 5
2 F ∴ .x > 0 1, 5
F 7 3
sin x + cos x = a (1)
sin x − cos x = b (2)
(1) + (2), 2 sin x = a + b
(1) − (2), 2 cos x = a − b
(1) × (2),sin2x − cos2x = ab → cos2x − sin2x = − ab
∴ =sin 4x sin 2(2x)
= 2 sin 2x cos 2x
= 2(2 sin x cos x)(cos2x − sin2x)
= (a + b)(a − b)(−ab) = (a2 − b2)(−ab) = ab3 − a3b
-2 1 2 1
3
ˆ F F F

16. F 8 3
ก 25x2 + 21y2 + 100x − 42y − 404 = 0
ก F = 40425x2 + 100x + 21y2 − 42y
=25(x2 + 4x + 4) + 21(y2 − 2y + 1) 404 + 100 + 21
= 52525(x + 2)2 + 21(y − 1)2
= 1(x +2)2
21
+
(y−1)2
25
F ก c = = 25 − 21 = 2
F HYPER ก F (−3,1 + 8 )
HYPER ก F
ก HYPER (y −1)2
22
−
(x+2)2
b2
= 1
HYPER F (−3,1 + 8 )
F (1 + 8 −1)2
4
−
(−3+ 2)2
b2
= 1
2 − 1
b2
= 1 → b2 = 1
ก HYPER (y −1)2
4
−
(x+ 2)2
1
= 1
4 F (y − 1)2
− 4(x + 2)2 = 4
y2 − 2y + 1 − 4(x2 + 4x + 4) = 4
y2 − 2y + 1 − 4x2 − 16x − 16 = 4
y2 − 4x2 − 2y − 16x − 19 = 0
ก - F
F
(-2,1)
c = 2 = aHYPER
y'
x'
F1
F2
(-3,1 + 8)
4
ˆ F F F

17. F 9 4
ก 1 ก mAB = 5− 1
1− (−3)
= 1 mDC = −3− 3
2− 8
= 1
F AB ก F DCmAB = mDC
ก 2 ก AB = [1 − (−3)]2 + (5 − 1)2 = 32 = 4 2
DC = (8 − 2)2 + [3 − (−3)]2 = 72 = 6 2
∴ AB + DC = 4 2 + 6 2 = 10 2
ก 3 ก ก ˈmCD = 1 CD x − y − 5 = 0
ก ก A CD
−3 −1−5
2
= 9
2
= 9
2
⋅
2
2
=
9 2
2
ก 4 ก ก B CD
1 −5− 5
2
= 9
2
ก F ABCD
ก F F
ก A ก BCD = CD
F 4
F 10 1
ก ก2y = b logy2x = a
log22y = log2b 2x = ya
∴ y = log2b x = 1
2
ya
Fy = log2b x = 1
2
(log2b)a
D
A B
C
5
ˆ F F F

18. F 11 2
72x + 72 < 23(x +1) + 32(x +1)
9x ⋅ 8x + 72 − 8x+1 − 9x +1 < 0
9x8x − 8x ⋅ 8 − 9x ⋅ 9 + 72 < 0
8x(9x − 8) − 9(9x − 8) < 0
(9x − 8)(8x − 9) < 0
F 9x − 8 = 0 → 9x = 8 → log99x = log98 → x = log98
F 8x − 9 = 0 → 8x = 9 → log88x = log89 → x = log89
ก (9x − 8)(8x − 9) < 0 (log98, log89)
F 12 2


1
4


x
+ 

1
2


x −1
+ a = 0 → 

1
2


2x
+ 2

1
2


x
+ a = 0


1
2


2x
+ 2

1
2


x
+ 1 = 1 − a → 



1
2


x
+ 1


2
= 1 − a
กก F x ∈ R+
0 < 

1
2


x
< 1 1 < 1 − a < 4
1 < 

1
2


x
+ 1 < 2 0 < − a < 3
1 < 



1
2


x
+ 1


2
< 4 0 > a > − 3
∴ a ∈ (−3,0)
ก ∴ F F 3, 4

1
4


x
+ 

1
2


x −1
+ a = 0 a < 0
ˈ ก F
F 1 ∴ F 2x = 1, 1
4
+ 1 + a = 0 → a = − 5
4
log 89 log 98
y
(0,1) y = ( )x1
2
x
6
ˆ F F F

19. F 13 1
x
x −1
= sec2θ
x = sec2θx − sec2θ
sec2θ = sec2θ ⋅ x − x
sec2θ = x(sec2θ − 1)
x = sec2θ
sec2θ−1
∴ =f(sec2θ) 1
sec2θ
sec2θ−1
= sec2θ− 1
sec2θ
= 1 − 1
sec2θ
= 1 − cos2θ = sin2θ
F 14 2
ก = 3 F กก F = 0b a b a ⋅ b
= 3 = 0(−2p)2 + 22 + p2 (1)(−2p) + 

1
2

 (2) + (−3p)(p)
= 3 = 05p2 + 4 −2p + 1 − 3p2
กก F = 03p2 + 2p − 1
F = 9 = 05p2 + 4 3(1) + 2p − 1
= 5 p =5p2 −1
= 1p2
∴ F p F กก F Fa b b = 3 −1 
−3
2
,0

F 15 1
ก F Fก F F
ก mAB = 2− 0
2− 0
= 1
กก FCD AB mCD = − 1
ก mCD =
y− 1
x− 1
F −1 =
y −1
x −1
−1(x − 1) = y − 1
−x + 1 = y − 1
F y = 2 − x (1)
y
x
C(x,y)
B(2,2)
D(1,1)
A(0,0)
7
ˆ F F F

20. ก F∆ABC = 4 1
2
(AB)(CD) = 4
1
2
(2 2 )CD = 4 → CD = 2 2
ก CD = (x − 1)2 + (y − 1)2
2 2 = (x − 1)2 + (2 − x − 1)2
2 2 = 2(x − 1)2
2 2 = 2 x − 1
2 = x − 1 → x = − 1,3
F (1) Fx = − 1 y = 2 − (−1) = 3
∴ C ก ˈ (−1,3)
C ก F Choice F ก Choice 1 ˈ
F 16 2
z1 = 0
n = 1, z2 = z1
2
+ i = i
n = 2, z3 = z2
2
+ i = i2 + i = − 1 + i
n = 3, z4 = z3
2
+ i = (−1 + i)2 + i = − 2i + i = − i
n = 4, z5 = z4
2
+ i = (−i)2 + i = − 1 + i
n = 5, z6
2
= (−1 + i)2 + i = − i
∴ z111 = − 1 + i = 2
F 17 4
S∞ =
n = 1
∞
Σ an =
n = 1
∞
Σ 

3n +2n − 2
4n−1

 =
n = 1
∞
Σ 

3n
4n−1
+ 2n
4n− 1
− 2
4n −1


S∞ =
n = 1
∞
Σ 3n
4n −1
+
n = 1
∞
Σ 2n
4n−1
−
n = 1
∞
Σ 2
4n −1
S∞ = 
3 + 9
4
+ 27
16
+ .....
 + 
2 + 1 + 1
2
+ .....
 − 
2 + 1
2
+ 1
8
+ .....

S∞ = 3
1− 3
4
+ 2
1 − 1
2
− 2
1− 1
4
= 12 + 4 − 8
3
= 40
3
3 F F F
C F Q2
8
ˆ F F F

21. F 18 2
ก f(x) = 3x
2
3
∴f (x) = 2x
−1
3 f (8) = 2(23)
−1
3 = 2

1
2

 = 1
F =(fog) (1) f (g(1)) ⋅ g (1)
= f (8) ⋅ 

2
3


= (1)

2
3

 = 2
3
F 19 1
ก F 13 4 13 × 4 = 52
n(S) = F 3 ก 52 F 

52
3

 = 22100
n(E) = F 3 F ก 2 F


13
2




2
1




4
2




4
1

 = 3744
ก 2 2 2
OR 

13
1




4
2




48
1

 = 3744
2 2
P(E) =
n(E)
n(S)
= 3744
22100
= 72
425
F 20 2
ก. ก F
n(A) = n(A ∩ B) + n(A ∩ B )
F
P(A) = P(A ∩ B) + P(A ∩ B )
ก. ก
. ก F
.P(A − B) = 0.2
A B
'A B
⊂
A B
⊂
A B
' 'P(A B )
⊂
0.2
0.2 0.3 0.3
9
ˆ F F F

22. F 21 3
ก =µ
N1µ1 +N2µ2
N1 +N2
F 40 = 35N +50N
N +N
40N + 40N = 35N 50N+
50N 10N=
=N
N
10
5
= 2
1
F 22 3
A = 7(77)
F ∴ F 4A > B
B = 777 = (711)7
F ∴ F 1, 2B > C
∴ F 3C = 777
F 23 3
กก ก PAT
1. ก 5 ก ˈ 17
2. F ก , F, , F,
3. F ก F ก
3 F F F ˈ F 3
F 24 4
F ก. F ก.a ∗ b = ab, b ∗ a = ba ab ≠ ba
F . (a ∗ b) ∗ c = (ab) ∗ c = (ab)c = abc
a ∗ (b ∗ c) = a ∗ (bc) = abc
F . a ∗ (b + c) = a(b +c)
(a ∗ b) + (a ∗ c) = ab + ac
F . (a + b) ∗ c = (a + b)c
(a ∗ c) + (b ∗ c) = ac + bc
F .
F .
F .
=/
=/
=/
10
ˆ F F F

23. F 25 1
1. F A C ก ก D
D ก ก F F A ก ก
2. A ก ก F B ก ก
3. B ก ก F E
4. E F D ก ก
5. D ก ก F C
F 26 18
ก F Fก F
F - F F
n(A ∩ B ∩ C ) = 11 n((B − A) ∩ (B − C)) = 15
A ∩ B ∩ C n((A ∩ B) ∪ (A ∩ C) ∪ (B ∩ C)) = 47
= F
∴ n(A ∩ B ∩ C) = n(A ∪ B ∪ C) − n(A ∪ B) = 91 − 11 − 15 − 47 = 18
F 27 5
กก 2 (3x + 1) + 2 3x + 1 ⋅ x − 1 + x − 1 = 7x + 1
2 3x + 1 ⋅ x − 1 = 3x + 1
กก 2 4(3x + 1)(x − 1) = (3x + 1)2
0 = (3x + 1)2 − 4(3x + 1)(x − 1)
0 = (3x + 1) ⋅ [3x + 1 − 4(x − 1)]
0 = (3x + 1)(−x + 5)
∴ x = − 1
3
,5
F F F Fx = − 1
3
( −1
3
− 1 ∉ R)
F Fx = 5 ( 16 + 4 = 36 )
A B
C
11
ˆ F F F

24. F 28 25
A = {2, 3, 5, 7} B = {1, 2, 3, ....., 10}
A B
2 1 ก(a,f(a)) ≠ 1 a ∈ A
3 2 A B
5 3 2 → 2, 4, 6, 8, 10
7 .
.. 3 → 3, 6, 9
10 5 → 5, 10
7 → 7
1 7 F Ff(7) = 7 (7,7) = 7 ≠ 1
2 5 F 10 Ff(5) = 5 (5,5) = 5 ≠ 1
(5,10) = 5 ≠ 1
3 3 F 6 9 Ff(3) = 3
(3,3) = 3 ≠ 1,(3,6) = 3 ≠ 1 (3,9) = 3 ≠ 1
4 2 F ˈ
ก 1 F Ff(5) ≠ 10 f(3) ≠ 6 f(2) = 2,4,6,8,10
ก 2 F Ff(5) ≠ 10 f(3) = 6 f(2) = 2,4,8,10
ก 3 F Ff(5) = 10 f(3) ≠ 6 f(2) = 2,4,6,8
ก 4 F Ff(5) = 10 f(3) = 6 f(2) = 2,4,8
12
ˆ F F F

25. ∴ ก F F 25
7 7
5 5
3 3
3 3
2 2 (1)
2 2 (10)
2 2 (15)
2 2 (22)
2 2 (19)
2 2 (6)
2 4 (2)
2 4 (11)
2 4 (16)
2 4 (23)
2 4 (20)
2 4 (7)
2 6 (3)
2 6 (12)
2 6 (17)
2 6 (24)
2 8 (4)
2 8 (13)
2 8 (18)
2 8 (25)
2 8 (21)
2 8 (8)
2 10 (5)
2 10 (9)
2 10 (14)
3 6
3 6
3 9
3 9
5 10
13
ˆ F F F

26. F 29 1
2
a2 + b2 sin α = a
a2 +b2
, cos β = a
a2 +b2
cos [arcsin (sin α)] + sin [arccos (cos β)] = 1
cos α + sin β = 1
cos (90 − β) + sin β = 1
∴sin β + sin β = 1 sin β = 1
2
F 30 1
2
= sin54 − sin 18
2 sin18 cos18 sin18
cos 18
+1 −2sin218
= 2 cos 36 sin 18
= 2 sin18 cos18 cos36
cos 18
= 2 sin36 cos36
2cos 18
= sin 72
2sin72
= 1
2
F 31 32
2A − B =



−4 −4
5 6


 (1)
A − 2B =



−5 −8
4 0


 (2)
(1) × 2 : 4A − 2B =



−8 −8
10 12


 (3)
(3) − (2) : 3A =



−3 0
6 12


 → A = 1
3



−3 0
6 12


 =



−1 0
2 4



det A =
−1 0
2 4
= − 4
A (1) F 2



−1 0
2 4


 − B =



−4 −4
5 6



B =



−2 0
4 8


 −



−4 −4
5 6


 =



2 4
−1 2



det B =
2 4
−1 2
= 8
= =det(A4B−1) (det A4)(det B−1) = (det A)4

1
detB

 (−4)4

1
8

 = 32
a
b
β
α
0
4
4
-4
14
ˆ F F F

27. F 32 6



1 0
−1 w






x −1
0 y


 =



2y −1
z 2






1 0
−1 w






x −1
−x 1 + yw


 =



2y + 1 −w
z − 2 2w



ก 1 ก 2
F −1 = − w → w = 1
ก 2 ก 2
F 1 + yw = 2w → 1 + y(1) = 2(1) → y = 1
ก 1 ก 1
F x = 2y + 1 = 2(1) + 1 = 3 → x = 3
ก 2 ก 1
F −x = z − 2 → − 3 = z − 2 → z = −1
∴ F 4w − 3z + 2y − x = 4(1) − 3(−1) + 2(1) − 3 = 6
F 33 3
ก F Fu ⋅ w = 2 1a + 2b + 3c = 2 (1)
ก กw = ai + bj + ck −2
3
i + 1
2
j + 1
3
k
F m ˈ F





a
b
c





= m







−2
3
1
2
1
3







a = m
−2
3

 m = a
−2
3
= b
1
2
= c
1
3
F Fb = m

1
2

 −3a
2
= 2b = 3c (2)
c = m

1
3


F 2b 3c ก (2) (1)
F 1a + 
−3a
2

 + 
−3a
2

 = 2 → a = −1
F a (2) F 3
2
= 2b = 3c → b = 3
4
,c = 1
2
∴a + 4b + 2c = − 1 + 4

3
4

 + 2

1
2

 = 3
15
ˆ F F F

28. F 34 5
z2 = 1 + 2i ,z2 = 1 − 2i
5z1 + 2z2 = 5
5z1 + 2(1 − 2i) = 5
5z1 = 3 + 4i
5 z1 = 3 + 4i → z1 = 1
∴ 5z−1 = 5
z = 5
z
= 5
1
= 5
F 35 1
an =
n
2
(2+2n)
n2
= n2 +n
n2
∴ n→ ∞
lim an =
n→ ∞
lim n2 +n
n2
= 1
F 36 1
=1
k (k+1)+ k k+ 1
1
k k +1



1
k +1 + k



k+1 − k
( k+1 − k )
= 1
k k +1
( k + 1 − k )
= 1
k
− 1
k +1
Sn =
k = 1
n
Σ



1
k
− 1
k+1


 = 1 − 1
2
+ 1
2
− 1
3
+ 1
3
− 1
4
+ ..... + 1
n
− 1
n +1
∴n→ ∞
lim Sn =
n →∞
lim


1 − 1
n +1


 = 1
F 37 53
Con xxxx ==== 2222
f(2) = a − b
x→ 2−
lim f(x) =
x→ 2−
lim x3 −3x −2
x −2
=
x→ 2−
lim 3x2 − 3
1
= 9
x→ 2+
lim f(x) =
x→ 2+
lim x2 + ax + 1 = 2a + 5
2a + 5 = 9 → a = 2
a − b = 9 → 2 − b = 9 → b = − 7
∴ a2 + b2 = 4 + 49 = 53
=
16
ˆ F F F

29. F 38 6
f(x) = ∫ f (x)dx = ∫(3x
1
2 + 5)dx = 3x
3
2
3
2
+ 5x + c
f(x) = 2x
3
2 + 5x + c
f(1) = 2 + 5 + c = 5 → c = − 2
∴f(x) = 2x
3
2 + 5x − 2
f(x2) = 2x3 + 5x2 − 2
∴ =
x→ 4
lim
f(x2)− 2
f(x) x→ 4
lim
(2x3 + 5x2 −2)− 2
2x
3
2 +5x− 2
= 128+ 80−4
16+20− 2
= 204
34
= 6
F 39 7
ก F F F (2, 19) F ก 19y = f(x)
F f (2) = 19 f(2) = 19
ก f (x) = 6x + 4
f (x) = ∫ (6x + 4)dx = 3x2 + 4x + c
∴f (2) = 12 + 8 + c = 19 → c = − 1 f (x) = 3x2 + 4x − 1
f(x) = ∫(3x2 + 4x − 1)dx = x3 + 2x2 − x + c
f(2) = 8 + 8 − 2 + c = 19 → c = 5
∴ f(x) = x3 + 2x2 − x + 5
f(1) = 1 + 2 − 1 + 5 = 7
F 40 44
ก 3 ก = 242 × 4 × 3
ก 2 ก = 164 × 4
ก 1 ก = 44
44
1,2
17
ˆ F F F

30. F 41 192
ก F F 4!2!4 = 192
F 42 520
ก µ = Σx
N
F = 72 N72 =
Σ x
N
→ Σx
70 =
Σ x + 60
N+ 1
+ 6070N + 70 = Σx
70N + 70 = 72N + 60
N = 5
ก σ2 = Σx2
N
− µ2
= 28920600 =
Σ x2
5
− 722 → Σx2
Fσ2 = 28920 +602
6
− 702
= 520
F 43 6
กก F F F
45 45 47 51
Med = 46
=µ 45 +45+47 +51
4
= 47
=σ2 Σ(x −µ)2
N
= 4 +4+ 0+16
4
= 6
18
ˆ F F F

31. F 44 10
ก z = x −µ
σ
F 4 = 700− µ
σ (1)
=−2
400− µ
σ (2)
6 =(1) − (2) 300
σ
= 50σ
= 500µ
F ก = σ
µ = 50
500
× 100 = 10%
F 45 7
31 F 3 F ก ก 10
F ˈ ก
ก 1 : 1 ก F
F F ก F F
1 2 3 4 5 6 7
29 30 31
ก ˈ F F F 5
กก ก 1 ก F ก ˈ F F F ก
ก 2 : 1 ก
F F ก F F
1 2 3 4 5
6 7 8 9 10 11 12
13 14 15 16 17 18 19
20 21 22 23 24 25 26
27 28 29 30 31
F ก ˈ F ก ก F ก F F 4
∴ 20 ก F
19
ˆ F F F

32. F 46 37
ก F ก = . . . 221 ก 260
. . . 221 ก 260 13221 = 13 × 17
∴ F ก ˈ ก 13 ก260 = 13 × 20
F ก 17 ก , 20 ก
∴ F F 37 ก
F 47 7
=(f ⊗ g)(1) f(g(1)) − g(f(1))
= f(3) − g(0)
= 8 − 1 = 7
F 48 b = 0
abcd ∴ a = 1 d = 9
9 ( F F ก 4 ก)a > 1 abcd × 9
dcba
ก 9cb 1 = 9 × (1bc9)
=9001 + 100c + 10b 9(1009 + 100b + 10c)
=9001 + 100c + 10b 9081 + 900b + 90c
= 8010c − 890b
∴ c = 8 + 89b (1)
c, b ˈ F F F0 → 9 b = 0 c = 8
F F ก (1) ˈ ( ก F F Fb > 0 c > 9
b = 1 → c = 97)
F 49
ก 6 + ก 5 = 1 + 2 + 3 + ..... + 11
43 + 28 − x = 11
2
(11 + 1) = 66
x = 5
20
ˆ F F F

33. F 50
I. 2 F 2 10 F 2
3 F 3 11 F 3
4 F 4 12 F 4
5 F 5 13 F 5
6 F 4 14 F 4
7 F 3 15 F 3
8 F 2 16 F 2
9 F 1 17 F 1
2400 F 8 ∴ 2400 F 2
II. 8 ก2400 = 8 × 300
∴ 2400 F 2
*************************
21
ˆ F F F